Functional quantization-based stratified sampling methods

TL;DR

This paper introduces quantization-based stratified sampling methods to effectively reduce variance in Monte Carlo simulations, with theoretical analysis and practical algorithms for Gaussian processes.

Contribution

It establishes a theoretical link between optimal quadratic quantization and variance reduction, and proposes new stratified sampling algorithms based on quantization techniques.

Findings

Quantization-based stratification achieves uniform efficiency for Lipschitz functionals.

The proposed methods effectively reduce variance in path-dependent functional simulations.

The balance between complexity and variance reduction is analyzed in detail.

Abstract

In this article, we propose several quantization-based stratified sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of stratification lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with stratified sampling. We first put the emphasis on the consistency of quantization for partitioning the state space in stratified sampling methods in both finite and infinite dimensional cases. We show that the proposed quantization-based strata design has uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein-Uhlenbeck processes. We derive in detail the case of Ornstein-Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction factor